In November/December 2020 Home Assistant went into a unhealthy state if Watchtower was found on the host.
It doesn't have to be a running container. It's enough to have a docker image on the host with a tag that starts with (or after the last slash, starts with) watchtower
.
Supervisor searches for images that matches a few patterns, and if an image matches, the system is marked as unhealthy and you cannot perform any upgrades.
If the Supervisor logs (Frontend > Supervisor > System) reports that you have containers that are not supported, you need to remove the images from the host.